The Joy of Math, or Fermat’s Revenge

For one brief shining moment,
it appeared as if the 20th century had justified itself. The era of world
wars, atom bombs, toxic waste, AIDS, Muzak and now, just to rub it in,
a pending Bush-Dukakis race, had redeemed itself, it seemed. It had brought
forth a miracle. Fermat’s last theorem had been solved.

Fermat’s last theorem is the world’s most famous unsolved mathematical
puzzle. It owes its fame to its age — it was born about five years before
Isaac Newton — and its simplicity. It consists of only one line. The Greeks
had shown that there are whole numbers for which a2+b2=c2. One solution for Pythagoras’ theorem, for example,
is 32+42=52. Pierre de Fermat conjectured
that the Pythagorean equation doesn’t work for higher dimensions: for n
greater
than 2, an+bn=cn is impossible.
It won’t work for n=3. (There are no integers for which
a3+b3=c3.)
Nor, theorized Fermat, for any higher
power: for n=4 or n=5 and so on.

Then came the mischief. Fermat left the following marginal annotation:
“I have discovered a truly remarkable proof [of this theorem], which this
margin is too small to contain.” And which for more than three centuries
the mind of man has been too dim to discern.

All these years mathematicians have given Fermat the benefit of the
doubt: the consensus was that the last theorem was probably true, but that
Fermat was mistaken in thinking or perverse in claiming that he had proved
it. Its legend grew as it defied 15 generations of the world’s greatest
mathematical minds. It became the Holy Grail of number theory. Then last
month came news that a 38-year-old Japanese assistant professor had found
the solution. Between the banal and the absurd that is the everyday, it
seemed, something epic had happened.

Alas, it had not. Yoichi Miyaoka and his colleagues have been checking,
and found fundamental if subtle problems deep in his proof. Miyaoka got
a glimpse of the Grail, but no more. The disappointment is keen — the 20th
century stands unredeemed — but it is mixed with a curious relief. “Next
to a battle lost,” wrote Wellington, “the greatest misery is a battle gained.”
Easy for him to say. (He won.) Still, there is wisdom in Wellington and
comfort too. Solving Fermat would have meant losing him. With Miyaoka’s
miss, Fermat — bemused, beguiling, daring posterity to best him — endures.

And mathematics gains. Miyaoka’s assault on Fermat is a reminder, an
enactment of the romance that is mathematics. Math has a bad name these
days. In the popular mind, it has become either a syndrome (math anxiety
is an affliction to be treated like fear of flying) or a mere skill. We
think of a math whiz as someone who can do in his head what a calculator
can do on silicon. But that is not math. That is accounting. Real math
is not crunching numbers but contemplating them and the mystery of their
connections. For Gauss, “higher arithmetic” was an “inexhaustible store
of interesting truths” about the magical relationship between sovereign
numbers. Real math is about whether Fermat was right.

Does it matter? It is the pride of political thought that ideas have
consequences. Mathematics, to its glory, is ideas without consequences.
“A mathematician,” says Paul Erdös, one of its greatest living practitioners
and one of the most eccentric, “is a machine for turning coffee into theorems.”
Mathematicians do not like to admit that, because when they do, their grant
money dries up — it is hard to export theorems — and they are suspected
of just playing around, which of course they are.

Politicians and journalists need to believe that everything ultimately
has a use and an application. So when a solution for something like Fermat’s
last theorem is announced, one hears that the proof may have some benefit
in the fields of, say, cryptography and computers. Mathematicians and their
sympathizers, at a loss to justify their existence, will be heard to say,
as a last resort, that doing mathematics is useful because “it sharpens
the mind.”

Sharpens the mind? For what? For figuring polling results or fathoming
Fellini movies or fixing shuttle boosters? We have our means and ends reversed.
What could be more important than divining the Absolute? “God made the
integers,” said a 19th century mathematician. “All the rest is the work
of man.” That work is mathematics, and that it should have to justify itself
by its applications, as a tool for making the mundane or improving the
ephemeral, is an affront not just to mathematics but to the creature that
invented it.

What higher calling can there be than searching for useless and beautiful
truths? Number theory is as beautiful and no more useless than mastery
of the balance beam or the well-thrown forward pass. And our culture expends
enormous sums on those exercises without asking what higher end they serve.

Moreover, of all such exercises, mathematics is the most sublime. It
is the metaphysics of modern man. It operates very close to religion, which
is why numerology is important to so many faiths and why a sense of the
transcendent is so keenly developed in many mathematicians, even the most
irreligious. Erdös, an agnostic, likes to speak of God’s having a
Book that contains the most elegant, most perfect mathematical proofs.
Erdös’ highest compliment, reports Paul Hoffman in the Atlantic,
is
that a proof is “straight from the Book.” Says Erdös: “You don’t have
to believe in God, but you should believe in the Book.”

In one of Borges’ short stories, a celestial librarian spends his entire
life vainly searching for a similar volume, the divine “total book” that
will explain the mystery of the universe. Then, realizing that such joy
is destined not to be his, he expresses the touching hope that it may at
least be someone else’s: “I pray to the unknown Gods that a man — just
one, even though it were thousands of years ago! — may have examined and
read it. If honor and wisdom and happiness are not for me, let them be
for others.”

For a couple of days it seemed that honor and wisdom and happiness were
Miyaoka’s. A mirage, it turns out. Yet someday Fermat’s last theorem will
be solved. You and I will not understand that perfect proof any more than
we understand Miyaoka’s version. Nonetheless, the thought that someone,
somewhere, someday, will be allowed a look at Fermat’s page in the Book
is for me, for now, joy enough.

Note from Joe: This article predates the now-famous proof of
Fermat’s Last Theorem by Andrew Wiles, but the points made by Krauthammer about mathematics
remain true, and this remains my all-time favorite magazine essay.